# Some question on the paper “Ricci-flat metrics on the complexification of a compact rank one symmetric space”

I am reading the paper "Ricci-flat metrics on the complexification of a compact rank one symmetric space" by Stenzel (here is the link:"http://www.math.osu.edu/~stenzel.3/research/publications/ricci-flat.pdf") and I have some questions.

1. On page 2 he makes the assumption that the linear isotropy group $Ad_{G}(K)$ acts transitively on the unit sphere in $\mathfrak{p}$. Why does he make this assumption? Why is this assumption good for the further proof?

2. On page 3 (on top) he says that "if $G/K$ is a compact, rank one, globally symmetric space, the orbits of $G$ in the Stein manifold $G_{\mathbb{C}} / K_{\mathbb{C}}$ are very easy to describe: they are hypersurfaces diffeomorphic to the sphere bundle in $T^{*}G/K$, and an exceptional orbit diffeomorpic to $G/K$.". Why is this so? Is there any reference?

3. On page 5 there is Theorem 3. The proof of Theorem 3 is in the preprint "An adapted complex structure on the cotangent bundle of a compact Riemannian homogeneous space". Where is this preprint? I cant find it. Can someone give a link or a similar reference where there is a proof?

4. Why do we need for this proof rank one ? What happens in higher rank ?

I hope that someone could answer some of these questions. Thanks. bernard

-
I think you have not read Stenzel's paper carefully. For example, right after he makes his assumption on page 2, he explicitly explains that this is because he wants to consider rank 1 symmetric spaces and he explains that this is because then the problem he is trying to solve can be reduced to an ordinary differential equation. It is because this strategy fails in higher rank that his proof doesn't work in that case (which answers your question 4; also he points out that others have now solved the higher rank case). For Point 2, which is rather easy, check Helgason. About 3, ask the author. –  Robert Bryant Apr 10 '13 at 17:25
You mean the book "Differential Geometry, Lie Groups and Symmetric Spaces" by Helgason ? –  bernard Apr 11 '13 at 7:36
@bernard: Yes, that's most likely where it is. It might be in his 'Groups and Geometric Analysis' book, though. I don't remember and don't have it handy. However, it is most likely in the Matsushima reference in Stenzel's paper and also in the Patrizio & Wong reference. In either case, it isn't hard; it follows from the way $G$ as a real form sits in $G_\mathbb{C}$. –  Robert Bryant Apr 11 '13 at 12:35